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Intercepts and Asymptotes of Tangent Functions - Concept
The tangent identity is tan(theta)=sin(theta)/cos(theta), which means that whenever sin(theta)=0, tan(theta)=0, and whenever cos(theta)=0, tan(theta) is undefined (dividing by zero). When the tangent function is zero, it crosses the x-axis. Therefore, to find the intercepts, find when sin(theta)=0. To find the vertical asymptotes determine when cos(theta)=0.
Sometimes a homework or problem will ask you about the intercepts and asymptotes of a tangent function. Well let's investigate that. We start with the identity tangent theta equals sine theta over cosine theta. Let's first find the zeroes of tangent. Because of that identity, the zeroes of tangent will be exactly the same as the zeroes of sine. That is tangent theta equals zero and sine theta equals zero. Now sine theta equals zero at the integer multiples of pi. So theta equals for example 0, pi 2pi and so on.
Now one way to say this a little more compactly, is to call it n pi where n is a n integer so integer multiples of pi. Now what does this have to do with x intercepts? Well, the zeros become x intercepts when you graph. So the x intercepts, I'll abbreviate this way would be 0 0, pi 0, 2 pi 0 and so on. these are the zeroes of tangent and of course the second coordinate of an x intercept is going to be 0. Those are the x intercepts. What about the asymptotes?
Well, tangent theta is undefined when cosine theta equals zero. Remember that identity. Cosine theta equals 0 when theta equals pi over 2 plus n pi. Again where n is any integer. So, this would be for example pi over 2, 3 pi over 2, 5 pi over 2 and so on. Now what does this have to do with asymptotes? Well these are the places where our tangent is going to be undefined. So there will be vertical asymptotes at these places. Vertical asymptotes are x equals pi over 2, 3 pi over 2, 5 pi over 2 and so on. And of course it goes in the negative direction too. x equals negative pi over 2,. negative 3 pi over 2, negative 5 pi over 2 and so on.
If you're ever asked about the domain of the tangent function, since these are the places where tangent's undefined you'd say all real numbers except these. If you're ever asked about the range of the tangent function this it's all real numbers. You can get any number out of the tangent function because remember tangent represents the slope of the terminal side of an angle on the unit circle, that can be anything.
So once again the zeroes of tangents and the x intercepts, integer multiples of pi. Tangent's undefined, vertical asymptotes these are pi over 2 plus integer multiples of pi.